Zorn's Lemma Application for Finding Maximal Submodule?Since $\bar L$ is a maximal element in $\mathscr{F}$ you have that for every other element $L \in \mathscr{F}$, that is every other submodule of $M$ such that $x \not \in L$ and $N \subseteq L$ you have that if $L \supseteq \bar L$ then $L = \bar L$. This is the definition of maximal element of a poset.